1.2 Background: Linear Algebra This project will require very basic linear algebra - no more than matrix multiplication and the tools to calculate the required work will be provided so you can focus on the math. Consider the following matrix equation: ay) = (3 31) (:)'(_11)=(g::r:ii) If we dene A := (3 31), :E' := (Z), and I: := (11) then we can write this expression as ALE 3. Now observe that this expression is functionally equivalent to the following: Thus there are two ways to write the function f (9:, y) - as a either a matrix equation, or as a typical multivariate function. Symbolab is a great tool if you are unfamiliar with matrix multiplication. Click here to see the matrix multiplication above done in symbolab. Also, while we can't graph f (:13, y) fully, we have some tools to analyze this function: link. The motivation behind this project comes from an important topic in linear algebra - solving systems of equations. For example, we might be interested in know for which values of x and y the following is true: 1 1 3$+y 2xy This is equivalent to asking when f (:13, y) = (0,0) or equivalently A53" 3: 6. 1.3 Background: The Gradient For a multivariate function h from R\" > R, we can construct the gradient of h, denoted Vh. When n = 2 (the only case we consider) the gradient of h : R2 > R is: a a 83' By For example, the gradient of m(x, y) = ln(:c3 + 'y) is 2 Vm= 3a: 1 1 x3+y x3+y Vh = Vh(x, y) = R: an+l = an _ \"v.9(an) 90 method of gradient descent is an algorithm that generates a sequence of points that, under the appropriate conditions, will converge to the location of a local minimum of the function 9. Give a short explanation for why this algorithm will nd the location of the minimum. You might mention what problems one might encounter. For example, what can go wrong with 17? Give as much detail as possible. It would be a great idea to include a picture to show what this process looks like. Here n is a step size, and we must start with an initial location of (1'0 = ($0). The (xii) Turning our attention back to the functions f (:13, y) and L(:L', y) = M f (:13, y)||2, explain why we can't use the method of gradient descent on f (3;, go), but why we can use it on Mac, :0)- (xiii) Our goal with this project is to nd the (37, y) pair(s) that make the function f (at, y) = (0, 0). Explain why the results of applying the method of gradient descent on L(x, y) gives us just that. (xiv) Use the results of the Python code to estimate a point (3,1)) that makes f (:13, y) = 0. Verify that your guess makes f (at, y) = O. (xv) (bonus) What would have changed if our function did not have a solution? What would have changed in the steps above and what would have remained the same